STNB 2016 (30th edition)
On fields of definition of $\mathbb{Q}$-curves and Sato-Tate groups of abelian surfaces
Presenters
Abstract
Let $A$ be an abelian surface defined over $\mathbb{Q}$ that is isogenous
over $\overline{\mathbb{Q}}$ to the square of an elliptic curve $E$. If $E$ does not have
complex multiplication (CM), one can deduce from results of Ribet and
Elkies, concerning fields of definition of $\mathbb{Q}$-curves, that $E$ admits a
model defined over a biquadratic extension of $\mathbb{Q}$. We will show that, in
our context, one can adapt Ribet's methods to treat the case in which
$E$ has CM. We find two applications of this analysis to the theory of
Sato-Tate groups of abelian surfaces: First, we show that 18 of the 34
existing Sato-Tate groups of abelian surfaces over $\mathbb{Q}$, only occur among
at most 51 $\overline{\mathbb{Q}}$-isogeny classes of abelian surfaces over $\mathbb{Q}$; Second, we
provide an answer to a question of Serre on the
existence of a number field over which abelian surfaces can be defined
realizing each of the 52 existing Sato-Tate groups of abelian surfaces
over number fields. This is an ongoing project with Xevi Guitart.
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